An N-dimensional analogue of Szegö's limit theorem
Abstract
Let T denote the unit circle in the complex plane, let φ ∈ L (T ), and for (a ,...,a ) ∈ (ℤ \0) let Λ be the rectangular lattice: Λ = {(m ,...,m ) ∈ ℤ : m ∈ [0,a )}. For each positive integer p define Λ = {(m ,...,m ) ∈ ℤ : m ∈ [0,pa )}. The Toeplitz matrix Τ (φ): l (Λ ) → l (Λ ) with symbol φ is defined by (equation presented) where {φ̂(m)} denotes the Fourier coefficients of φ. Assuming appropriate conditions on φ and on a function F we find N + 1 terms of the asymptotic expansion of the trace of F(Τ (φ)) as p → ∞. This expansion takes the form (equation presented) where the coefficients c = c (φ) depend on the N - J dimensional faces of Λ. We also find an expansion for the case when the edges of Λ expand at different rates and we apply this generalization to compute an example. ∞ N N N N 2 2 N 1 N + 1 N i i p 1 N i i p p p m∈ℤ p J,F J,F
